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Local Structure of Principally Polarized Stable Lagrangian Fibrations 3

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Local Structure of Principally Polarized Stable Lagrangian Fibrations 3 LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS JUN-MUK HWANG, KEIJI OGUISO Abstract. A holomorphic Lagrangian fibration is stable if the characteristic cycles of the singular fibers are of type Im, 1 ≤ m< ∞, or A∞. We will give a complete description of the local structure of a stable Lagrangian fibration when it is principally polarized. In particular, we give an explicit form of the period map of such a fibration and conversely, for a period map of the described type, we construct a principally polarized stable Lagrangian fibration with the given period map. This enables us to give a number of examples exhibiting interesting behavior of the characteristic cycles. 1. Introduction For a holomorphic symplectic manifold (M,ω), i.e., a 2n-dimensional complex manifold 0 2 with a holomorphic symplectic form ω H (M, ΩM ), a proper flat morphism f : M B over an n-dimensional complex manifold∈ B is called a (holomorphic) Lagrangian fibration→ if all smooth fibers are Lagrangian submanifolds of M. The discriminant D B, i.e., the set of critical values of f, is a hypersurface if it is non-empty. In [HO1], the⊂ structure of the singular fiber of f at a general point b D was studied. By introducing the notion of characteristic cycles, [HO1] shows that the∈ structure of such a singular fiber can be described in a manner completely parallel to Kodaira’s classification ([Kd], see also V. 7 in [BHPV]) of singular fibers of elliptic fibrations. Furthermore, to study the multiplicity of the singular fibers, [HO2] generalized the stable reduction theory of elliptic fibrations (cf. V.10 in [BHPV]), explicitly describing how arbitrary singular fiber over a general point of D can be transformed to a stable singular fiber, a singular fiber of particularly simple type. These results exhibit that the theory of general singular fibers of a holomorphic Lagrangian arXiv:1007.2043v1 [math.AG] 13 Jul 2010 fibration gives a very natural generalization of Kodaira’s theory of elliptic fibrations. The current work is yet another manifestation of this principle. An important part of Kodaira’s theory is the study of the asymptotic behavior of the elliptic modular function of a given elliptic fibration near a singular fiber. As a generalization of this we will study the asymptotic behavior of the periods of the abelian fibers near a general singular fiber of a holomorphic Lagrangian fibration. Here we need to make two additional assumptions on the Lagrangian fibration. First, we will assume that the singular fibers are of stable type, i.e., its characteristic cycles are of type Ik, 0 k (I meaning A ). As explained above, any general singular fiber can be transformed≤ ≤ ∞ into∞ this form by the∞ stable reduction ([HO2], Section 4). Jun-Muk Hwang is supported by National Researcher Program 2010-0020413 of NRF and MEST, and Keiji Oguiso is supported by JSPS Program 22340009 and by KIAS Scholar Program. 1 2 JUN-MUK HWANG, KEIJI OGUISO The second assumption we will make is that the Lagrangian fibration is principally po- larized, in the sense explained in Definition 3.1. This condition is satisfied if there exists 1 an f-ample line bundle on M f − (D) whose restriction on smooth fibers give principal polarizations on the abelian varieties.\ This assumption is rather restrictive compared with the setting of [HO1] and [HO2], where the only assumption was that the fibers of f are of Fujiki class. We believe that understanding the structure of Lagrangian fibration under these as- sumptions is essential for the study of general cases. Since this special case already requires substantial care and already provides many interesting examples (see Section 5), we restrict our attention to it in this paper and leave the general cases to future study. The main result of this paper is the following. Theorem 1.1. Let f : M B be a principally polarized stable Lagrangian fibration (cf. Definition 2.1 and Definition→ 3.1). Then at a general point b D of the discriminant, there ∈ exists a coordinate system (z1,...,zn) with D defined by zn = 0, such that for a suitable choice of an integral frame of the local system R1f Z on B D, the period matrices have the form ∗ \ 2 2 i ∂ Ψ n ∂ Ψ ℓ θj = for (i, j) = (n,n) and θn = + log zn ∂zi∂zj 6 ∂zn∂zn 2π√ 1 − where Ψ is a holomorphic function in z1,...,zn. Here ℓ is the number of irreducible com- ponents of a general singular fiber. i Conversely, given any germ of holomorphic function Ψ(z1,...,zn) such that Im(θj) > 0, there exists a principally polarized stable Lagrangian fibration whose period matrices are of the above form. That the period matrix of Lagrangian fibration is the Hessian of a potential function is a well-known consequence of the action-angle variables (cf. [DM]). The logarithmic behavior of the multi-valued part reflects the stability assumption on the singular fiber. The novelty in Theorem 1.1 lies in the choice of the variable zn through which these two aspects are intertwined. The existence of zn follows from the fact proved in Proposition 3.13 that the characteristic foliation accounts for the degenerate part of the polarization restricted to the fixed part of the monodromy. The proof of this uses a version of the Monodromy Theorem from the theory of the degeneration of Hodge structures and the topological property of the stable singular fiber. The converse direction in Theorem 1.1 is shown by explicitly constructing a principally polarized stable Lagrangian fibration from a given potential function Ψ(z) imaginary part of whose Hessian matrix is positive definite. This part is a sort of generalization of Nakamura’s construction [Na] of toroidal degeneration of principally polarized abelian varieties over 1- dimensional small disk. Using our construction, we shall give a concrete 4-dimensional example of principally polarized stable Lagrangian fibration in which the types of char- acteristic cycles of singular fibers change fiber by fiber, too. To our knowledge, such an example has not been noticed previously. In fact, most of the previous constructions of singular fibers of Lagrangian fibrations have used product construction from elliptic fibra- tions. LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 3 2. Stable Lagrangian fibrations Definition 2.1. A Lagrangian fibration is a proper flat morphism f : M B from a holomorphic symplectic manifold (M,ω) of dimension 2n to a complex manifold→ B of di- mension n such that the smooth locus of each fiber is a Lagrangian submanifold of M. The discriminant D B is the set of the critical values of f, which is a hypersurface in B if it is non-empty. Throughout⊂ this paper, we assume that D is non-empty. We say that f is a 1 stable Lagrangian fibration if D B is a submanifold and each singular fiber f − (b), b D, is stable, i.e., it is reduced and⊂ the characteristic cycle in the sense of [HO1] is of∈ type 1 Ik, 1 k . By the description in [HO1], this is equivalent to saying that f − (b) is reduced≤ and≤ ∞its normalization is a disjoint union of a finite number of compact complex manifolds Y 1,...,Y ℓ for some positive integer ℓ such that (i) each Y i is a P1-bundle over an (n 1)-dimensional complex torus Ai whose fibers − 1 are sent to characteristic leaves of f − (b) in the sense of [HO1], i.e., for a defining function h (B) of the divisor D, the Hamiltonian vector field ιω(f ∗dh), where 1 ∈ O ιω :ΩM T (M) is the vector bundle isomorphism induced by ω, is tangent to the image of→ the fibers in M; i i i i i (ii) there exist submanifolds S1,S2 Y , with S1 = S2 except possibly when ℓ = 1, 2, i i ⊂ 6 i 1 such that S1 S2 is a 2-to-1 unramified cover of A under the P -bundle projection; ∪ i 1 (iii) the normalization ν : Y f − (b) is obtained by the identification via a collection → i i+1 ℓ 1 of biholomorphic morphismsS gi : S2 S1 for 1 i ℓ 1 and gℓ : S2 S1 with 1→ 1 1 ≤ ≤2 − 2 → the additional requirement g1 = g2− if S1 = S2 and S1 = S2 for ℓ = 2. A maximal connected union of the P1-fibers in (i) under the identification in (iii) is called a characteristic cycle. A characteristic cycle can be either of finite type (Im-type, 1 m< ) or of infinite type A , which we also denote by I . ≤Recall∞ (cf. [HO2] Section 4) that∞ in a neighborhood of a gener∞ al singular fiber, any La- grangian fibration whose fibers are of Fujiki class can be transformed to a stable Lagrangian fibration by certain explicitly given bimeromorphic modifications and branched covering. We will be interested in the local property of the fibration at a point of D. Thus we will make the following (Assumption) D B is the germ of a smooth hypersurface in an n-dimensional complex manifold and the fundamental⊂ group π (B D) is cyclic. 1 \ The following is immediate from Proposition 2.2 of [HO1]. Proposition 2.2. Given a stable Lagrangian fibration, we can assume that there exists an n 1 action of the complex Lie group C − on M preserving the fibers and the symplectic form i i n 1 i such that S1,S2 are orbits of this action for all 1 i ℓ. This action of C − on Y i ≤ ≤ descends to the translation action on A .
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